A novel representation of the surface tension force for two-phase flow with reduced spurious currents E. Aulisa a , S. Manservisi a,b , R. Scardovelli b, * a Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042, USA b DIENCA, Laboratorio di Montecuccolino, Universita ` di Bologna, Via dei Colli 16, 40136 Bologna, Italy Received 1 June 2005; accepted 15 December 2005 Abstract A new numerical model for the solution of the single-fluid formulation of the incompressible Navier–Stokes equations for two-phase flows is presented. The model couples an interface front-tracking algorithm to a solver of the Navier–Stokes equations based on the finite element method. The interface line is described by an ordered set of markers which are advected along the flow streamlines and from the knowledge of their position the local normal and mean curvature are calculated. The local mean curvature has been extended to the whole computational domain and the singular surface tension force has been transformed within the framework of the variational for- mulation into a regular volume force. Isoparametric Taylor–Hood elements have been used to discretize the Navier–Stokes equations. The resulting system has been solved with a fully multigrid method coupled with a Vanka-like smoother. The model has been tested with a dynamical system characterized by a droplet surrounded by another fluid. Equilibrium solutions described by the Laplace’s law are reproduced with great accuracy with a very low level of spurious currents, which disappear completely if the exact value of the curvature is used. Damped oscillations are investigated in terms of the oscillation period, the damping coefficient and the critical Reynolds number for the transition from periodic to aperiodic regimes. Good agreement with theoretical solutions and the results of other numerical mod- els has been found when the comparison is applicable. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Navier–Stokes equations; Finite element methods; Interface front-tracking; Two-phase flows; Capillary force 1. Introduction Two-phase flows are of interest for many different applications ranging from environmental sciences to the oil and nuclear industries. In recent years this interest has inspired a great number of algorithms for the solution of these problems involving moving interfaces. Among them a partial list of some very popular approaches includes volume-of-fluid (VOF), level set, phase field and front tracking methods. An extensive number of numerical simulations has been performed in this area and the interested reader can consult the following articles [1–5] and references therein. The numerical methods can be divided into different groups depending on the type of discretization. In the first group moving grids are used and the interface is treated as a boundary, thus allowing a precise representation of interfacial jumps in the physical variables. The second group uses fixed grids and requires an interface representation and an advection algo- rithm in order to describe its motion across the computational grid. The first group of methods is very precise, but it can be used only in simulations with small deformations of the interface and may require regridding when the mesh becomes severely distorted or there are changes in the interface topology. The latter is widely used and it may suffer in some instances of a loss of accuracy and precision. 0045-7825/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2005.12.006 * Corresponding author. E-mail addresses: eugenio.aulisa@ttu.edu (E. Aulisa), sandro.manservisi@unibo.it (S. Manservisi), ruben.scardovelli@unibo.it (R. Scardovelli). www.elsevier.com/locate/cma Comput. Methods Appl. Mech. Engrg. 195 (2006) 6239–6257